# How do you differentiate #f(x) = sin((− x^2 − 1)^2) *(x^2 − 9)^2# using the chain rule?

Hey there!

To differentiate your function, you want to start by looking, overall, at what type of function you have. In short, you have a product (with chains inside). This could get a bit messy, haha!

Recall that chain rule looks like this:

If

First, start off with the product rule where if:

Note that the order of multiplication doesn't matter!

Starting off with your derivative:

The derivative of sin(x) is cos(x), but your "x" is much more complicated; it's a chain of functions and thus, by Chain Rule, you must multiply by the inside of the function. Therefore the derivative of the first part is:

So far we have:

Now we need to differentiate the second half of the equation. This is a simple chain rule problem.

By Chain Rule / Power Rule the derivative is:

Now you just multiply by the first part of the function unchanged to get:

Finally, combine everything to get:

I know this was a bit messy, but hopefully everything was clear! Chain rule, combined with the other rules for derivatives can get quite complicated, it just requires practice! Hopefully this helps! :)

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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